Engineering with Computers

, Volume 20, Issue 4, pp 342–353 | Cite as

Building spacetime meshes over arbitrary spatial domains

  • Jeff EricksonEmail author
  • Damrong Guoy
  • John M. Sullivan
  • Alper Üngör
Original article


We present an algorithm to construct meshes suitable for spacetime discontinuous Galerkin finite-element methods. Our method generalizes and improves the ‘Tent Pitcher’ algorithm of Üngör and Sheffer. Given an arbitrary simplicially meshed domain X of any dimension and a time interval [0, T], our algorithm builds a simplicial mesh of the spacetime domain X × [0, T], in constant time per element. Our algorithm avoids the limitations of previous methods by carefully adapting the durations of spacetime elements to the local quality and feature size of the underlying space mesh.


Wave Speed Discontinuous Galerkin Discontinuous Galerkin Method Space Mesh Cone Constraint 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The authors thank David Bunde, Michael Garland, Shripad Thite, and especially Bob Haber for several helpful comments and discussions. We also thank Shripad Thite for correcting a bug in our proof of Lemma 1. Jeff Erickson supported in part by a Sloan Fellowship and NSF CAREER award CCR-0093348. Damrong Gouy supported in part by DOE grant LLNL B341494. John M Sullivan supported in part by NSF grant DMS-00-71520. Alper Üngör This research was performed while this author was a student at the University of Illinois, with the additional support of a UIUC Computational Science and Engineering Fellowship.


  1. 1.
    Abedi R, Chung S-H, Erickson J, Fan Y, Garland M, Guoy D, Haber R, Sullivan JM, Thite S, Zhou Y (2004) Spacetime meshing with adaptive refinement and coarsening. In: Proceedings of the 20th annual ACM symposium on computational geometry, pp 300–309Google Scholar
  2. 2.
    Bern M, Chew LP, Eppstein D, Ruppert J (1995) Dihedral bounds for mesh generation in high dimensions. In: Proceedings of the 6th annual ACM-SIAM symposium on discrete algorithms, pp 89–196Google Scholar
  3. 3.
    Bern M, Eppstein D, Gilbert J (1994) Provably good mesh generation. J Comput Syst Sci 48:384–409Google Scholar
  4. 4.
    Cassidy C, Lord G (1980) A square acutely triangulated. J Rec Math 13(4):263–268Google Scholar
  5. 5.
    Cheng S-W, Dey TK, Edelsbrunner H, Facello MA, Teng S-H (1999) Sliver exudation. In: Proceedings of the 15th Annual ACM symposium on computational geometry, pp 1–13Google Scholar
  6. 6.
    Cockburn B, Karniadakis G, Shu C (2000) The development of discontinuous Galerkin methods. In: Discontinuous Galerkin methods: theory, computation and applications. Lecture notes in computer science engineering, vol 11. Springer, Berlin Heidelberg New York, pp 1–14Google Scholar
  7. 7.
    Cockburn B, Karniadakis G, Shu C (2000) Discontinuous Galerkin methods: theory, computation and applications. Lecture notes in computer science engineering, vol 11. Springer, Berlin Heidelberg New YorkGoogle Scholar
  8. 8.
    Eppstein D (1997) Acute square triangulation. The Geometry Junkyard. eppstein/junkyard/acute-square/
  9. 9.
    Erickson J, Gouy J, Sullivan J, Üngör A (2002) Building space–time meshes over arbitrary spatial domains. In: Proceedings of the 11th annual international meshing roundtable, pp 391–403.
  10. 10.
    Hangan T, Itoh J, (2000) Zamfirescu T Acute triangulations. Bull Math de la Soc des Sci Math de Roumanie 43:279–286MathSciNetGoogle Scholar
  11. 11.
    Lowrie RB, Roe PL, van Leer B (1998) Space–time methods for hyperbolic conservation laws. In: Barriers and challenges in computational fluid dynamics. ICASE/LaRC interdisciplinary series in science and engineering, vol 6. Kluwer, Dordrecht, pp 79–98Google Scholar
  12. 12.
    Maehara H (2000) On acute triangulations of quadrilaterals. In: Proceedings Japan Conference on Discrete Computational Geometry. Lecture notes in computer science, vol 2098. Springer, Berlin Heidelberg New York, pp 237–243.
  13. 13.
    Manheimer W (1960) Solution to problem E1406: dissecting an obtuse triangle into acute triangles. Amer Math Monthly 67:923Google Scholar
  14. 14.
    Richter GR (1994) An explicit finite element method for the wave equation. Appl Numer Math 16:65–80CrossRefGoogle Scholar
  15. 15.
    Ruppert J (1995) A Delaunay refinement algorithm for quality 2-dimensional mesh generation. J Algorithms 18(3):548–585CrossRefGoogle Scholar
  16. 16.
    Sheffer A, Üngör A, Teng S-H, Haber RB (2000) Generation of 2D space–time meshes obeying the cone constraint. Advances in computational engineering and sciences. Tech Science Press, Forsyth, pp 1360–1365Google Scholar
  17. 17.
    Shewchuk JR (1998) Tetrahedral mesh generation by Delaunay refinement. In: Proceedings of the 14th annual ACM symposium on computational geometry, pp 86–95Google Scholar
  18. 18.
    Thompson LL (1994) Design and analysis of space–time and Galerkin least-squares finite element methods for fluid-structure interaction in exterior domains. PhD Thesis, Stanford UniversityGoogle Scholar
  19. 19.
    Thite S (2004) Efficient spacetime meshing with nonlocal cone constraints. In: Proceedings of the 11th international meshing roundtable, pp 47–58. html
  20. 20.
    Üngör A (2001) Tiling 3D Euclidean space with acute tetrahedra. In: Proceedings of the 13th Canadian conference on computational geometry, pp 169–172.
  21. 21.
    Üngör A, Heeren C, Li X, Sheffer A, Haber RB, Teng S-H (2000) Constrained 2D space–time meshing with all tetrahedra. In: Proceedings of the 16th IMACS World CongressGoogle Scholar
  22. 22.
    Üngör A, Sheffer A (2000) A pitching tents in space–time: mesh generation for discontinuous Galerkin method. In: Proceedings of the 9th international meshing roundtable, pp 111–122.
  23. 23.
    Üngör A, Sheffer A, Haber RB (2000) Space–time meshes for nonlinear hyperbolic problems satisfying a nonuniform angle constraint. In: Proceedings of the 7th international conference on numerical grid generation in computational field simulationsGoogle Scholar
  24. 24.
    Üngör A, Sheffer A, Haber RB, Teng S-H (2002) Layer based solutions for constrained space-time meshing. To appear in Applied Numer Math.
  25. 25.
    Yin L, Acharya A, Sobh N, Haber R, Tortorelli DA (200) A space–time discontinuous Galerkin method for elastodynamic analysis. In: Discontinuous Galerkin methods: theory, computation and applications. Lecture notes in computer science engineering, vol 11. Springer, Berlin Heidelberg New York, pp 459–464Google Scholar

Copyright information

© Springer-Verlag London Limited 2005

Authors and Affiliations

  • Jeff Erickson
    • 1
    Email author
  • Damrong Guoy
    • 2
  • John M. Sullivan
    • 3
  • Alper Üngör
    • 4
  1. 1.Department of Computer ScienceUniversity of Illinois at Urbana-ChampaignUrbana-ChampaignUSA
  2. 2.Center for Simulation of Advanced Rockets, Computational Science and Engineering ProgramUniversity of Illinois at Urbana-ChampaignUrbana-ChampaignUSA
  3. 3.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbana-ChampaignUSA
  4. 4.Center for Geometric and Biological Computing, Department of Computer ScienceDuke UniversityDurhamUSA

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